From: Dan Asimov <dasimov@earthlink.net> Given arbitrary integers A, B, C they determine the quadratic form Q(x,y) = Ax^2 + Bxy + Cy^2 * Does the limiting density dens(A,B,C) of the set of integers represented by Q always exist??? * If so, can one determine its value??? Here dens(A,B,C) is defined as follows: Define the set Q_R := {Q(x,y) | (x,y) \in Z^2 with x^2+y^2 <= R^2} Note that any repeated value of Q(x,y) appears in Q_R with multiplicity = 1 here. Now let dens(A,B,C) := limit as R -> oo of (card(Q_R) / (pi R^2)), if it exists. (Here pi R^2 is a stand-in for the number of lattice points lying inside the disk of radius R about the origin. But the two expressions are asymptotic to each other, so this should not be a problem.)

Problem: Given any integers A, B, C does the limit dens(A,B,C) exist? And if so, what is its value?

--yes, this density always exists because its value always is always zero! More interesting question would be, the "density ratio" for Q versus some fixed forms such as x^2+y^2, x^2+xy+y^2, or, for indefinite forms, xy. Does that always exist, and if so what is its value. (I believe this answer is known to be "yes"... you'd need to search the literature.)